In category theory, two categories C and D are isomorphic if there exist functorsF : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.

Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of equivalence of categories; roughly speaking, for an equivalence of categories we don't require that be equal to , but only naturally isomorphic to , and likewise that be naturally isomorphic to .

if C is isomorphic to D and D is isomorphic to E, then C is isomorphic to E.

A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets. This criterion can be convenient as it avoids the need to construct the inverse functor G.

for every v in V and every element Σ agg in kG. Conversely, given a left kG module M, then M is a k vector space, and multiplication with an element g of G yields a k-linear automorphism of M (since g is invertible in kG), which describes a group homomorphism G → GL(M). (There are still several things to check: both these assignments are functors, i.e. they can be applied to maps between group representations resp. kG modules, and they are inverse to each other, both on objects and on morphisms).

Another isomorphism of categories arises in the theory of Boolean algebras: the category of Boolean algebras is isomorphic to the category of Boolean rings. Given a Boolean algebra B, we turn B into a Boolean ring by using the symmetric difference as addition and the meet operation as multiplication. Conversely, given a Boolean ring R, we define the join operation by ab = a + b + ab, and the meet operation as multiplication. Again, both of these assignments can be extended to morphisms to yield functors, and these functors are inverse to each other.

Further, if C is a category with an initial object s, then the slice category (s↓C) is isomorphic to C. Dually, if t is a terminal object in C, the functor category (C↓t) is isomorphic to C. Similarly, if 1 is the category with one object and only its identity morphism (in fact, 1 is the terminal category), and C is any category, then the functor category C1, with objects functors c: 1 → C, selecting an object c∈Ob(C), and arrows natural transformations f: c → d between these functors, selecting a morphism f: c → d in C, is again isomorphic to C.